Suggestions for objective function

[ivarne]

Currently Julia performs poorly with anonymous functions, and the sorting system has used a funny trick with types to pass a different comparison function. After reading @mauro3's comment about implicit objective functions in #30, I got an idea for a proposal for how to solve 3 problems in a somewhat cluncky way.

immutable Density <: ODEImplicit 
    mu::Float64
end

function ode_fun(typ::Density, y, y_t, t)
    return (typ.mu .* y .* t) - y_t
end

t_out, y_out = ode_xx(Density(1.091), y0, [0. 1.])

This would of be in addition to the normal F(y,t) = y', system currently proposed. This approach has 3 main advantages.

1. It uses types instead of anonymous functions. That enables the use of Julia's dispatch and inference system to generate specific code for each ODE problem. Better performance will probably be the result.
2. It gives a nice way to access system properties inside the differentiation function. If you do not need system properties, you can define the immutable/type as empty.
3. It gives a natural way to tag different problem specifications in a way that encourages multiple dispatch.

[ivarne]

Hmm... Different objective function specifications can actually just be controlled by a keyword argument. We would need to call a different internal solver function anyway, and then the check of the keyword argument will be outside the inner loop and not affect the types of the returned values.

[mauro3]

In 0.4, types can be called and have good type inference (as I'm sure @ivarne is aware of). I think it should be possible to use that out of the box with ODE.jl. Maybe it would make sense to "sanction" the use of the "Functors" of base: https://github.com/JuliaLang/julia/blob/0a341a6e7ece5802d45901797f304b85420e1eb2/base/functors.jl